Is there an addon or a trick to prevent objects from being placed inside of each other?
You can use Rigid Body simulation, maybe combined with some forces to distribute balloons. That may take some time and experiments to make it right.



You can automate that by mass copying rigid body properties from one balloon to others. Please check Blender docs to know more about Rigid Bodies and Force Fields.

You can automate your work by writing small python script. Please note that a lot of commands you need can be taken from the console.



Depending on how "big" number of balloons you want using a particle system for instancing balloons can be quite good solution. To make balloons more diverse, there can be made several versions of them and grouped. Then you just need to use that group for instancing and adjust settings like number, size variation, etc. Unfortunately Blender currently does not directly support Rigid Bodies for Particle System instances so they need to be converted to objects first.